On <span class="mathjax-formula">$\lambda (\phi ,P)$</span>-nuclearity

C ^OMPOSITIO M ^ATHEMATICA

M. S. R ^AMANUJAN B. R ^OSENBERGER

On λ(φ, P)-nuclearity

Compositio Mathematica, tome 34, n

^o

2 (1977), p. 113-125

<http://www.numdam.org/item?id=CM_1977__34_2_113_0>

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113

ON

03BB (03A6, P)-NUCLEARITY

M. S.

Ramanujan*

^{and B.}

Rosenberger

Noordhoff International Publishing

Printed in the Netherlands

Introduction

We consider _sequence

spaces 03BB (~, P, ~)

^and

03BB (~, P, k)

^{and define}

À (0, P, 00 )-nuclear

resp.

k (0, P, N)-nuclear

spaces in order to

unify

^the

concepts

^of

À(P)-nuclearity, 039BN(03B1)-nuclearity, and 0 -nuclearity

^con-

sidered in

[7,8,9] ;

^{we are}

especially

interested in

obtaining

^universal

generators ^{for the} varieties of

03BB (~, P, ~)-nuclear

^and

k (0, P, N)-

nuclear _spaces. Section 1 of the _paper contains various definitions and

some remarks on the operator ^{ideal of}

À(P, ~)-nuclear

maps, P ^a stable, ^nuclear

G--set;

in section 2 we consider

À(cp, P, (0)-

nuclear _spaces and show that

03BB (~, P, 00 )-nuclearity

^{is the same as}

À(Q, -)-nuclearity, Q

^a

suitably

chosen Gao-set. In section 3 we

introduce the

concept

^{of À}

(~, P, N)-nuclearity

and extend a result in

[8] by showing

^that

03BB (Q, ~) - Q suitably

chosen G--set - is a universal generator ^{for the}

variety

^of

03BB (~, P, N)-nuclear

spaces whenever P is a

countable,

monotone,

stable,

nuclear G--set.

1.

Definitions, notations,

^{and some remarks on}

A.

(P, 00 )-nuclear

maps

For

terminology

^and notations not

explained

^{here we refer to}

Kôthe

[3],

^Pietsch

[4], Dubinsky

^and

Ramanujan [1],

^and

Terzioglu [12].

Let X and Y be Banach

spaces, À

^{a normal} sequence space, and 03BB

* The first author acknowledges ^with pleasure ^the support of this work under SFB 72 at the University of Bonn and the hospitality of Professor E. Schock.

its Kôthe-dual. A continuous linear _{map T} E

L(X, Y)

is said to be

(i)

À -nuclear

(written

^{T E}

N03BB (X, Y))

if there exists a

representation

(ü) pseudo-A-nuclear

^or

À-nuclear (written

a

representation

(iii) of

type ^{À if}

{Snapp(T)}n

^{~ 03BB where}

s:PP(T)

^{denotes the n-th}

approximation

^{number of T.}

For a

locally

^{convex Hausdorff} space

(l.c.s.) E, U(E)

will denote a

neighbourhood

^{base of 0 of}

absolutely

convex, closed sets; Eu ^will denote the

completion

^of the normed space

ElpY(0)

^{U E}

OU(E);

8n(V, U)

^{denotes the n -th}

Kolmogorov-diameter

^{of V E}

U(E)

^with respect ^{to U e}

6U(E); L1(E)

denotes the â -diametral dimension of

E, viz.,

^the sequences

(yn)n

^{such that}

given U ~ U (E)

there exists a

V E

U (E)

^with

03B3n03B4n ( V, U ) ~ 0.

Let

03BB (P )

^{be a Kôthe} space with its

generating

^{Kôthe set P. The}

Kôthe set P is called a power ^set

of infinite

type if it satisfies the

following

additional conditions:

(i)

for each a E

P,

⁰ an - an+1, n E N;

(ü)

for each a E

P,

there exists a b e P with

ai = bn, n

^{E N.}

The

corresponding

space

03BB (P, ~)

^{is called a smooth} sequence space

of infinite

type ^{or a}

Gao-space.

^{For an}

example

^{of a}

G--space

^{which is not}

a power series _space of infinite

type

^see

[1;

^theorem

2.25].

Throughout, À(P, (0)

is assumed to be a

Gao-space.

^The

nuclearity

and related concepts ^{of such} spaces ^are discussed in

[1, 12,13] ;

^we

only

^{need the}

following

^result.

1.1

LEMMA: 03BB(P, ~)

is nuclear

if

^and

only if

^{there exists a} sequence

{Pn}n

^{E P such that}

{1/pn}n

^{E 11-}

We shall

frequently

say "P is a nuclear G--set" to mean that the

corresponding À (P, ~)

^{is a nuclear}

G~-space;

^{P is said to be a}

countable,

monotone, ^nuclear G--set if P ⁼

{{pin}n :

^{i =}

1,2, ...}

^with

p in ~ pri

^{for each}

i,

^{n E N and P is a nuclear}

Gao-set;

^{note that} P is a

G--set

already implies

^{0 ~}

p in ~ pin+1.

Let

À(P, ~)

^be

nuclear;

^{then a l.c.s. E is said to be}

À (P, ~)-nuclear

if for each U E

U (E)

there exists a V E

U (E)

such that V is absorbed

by

^U and the canonical _map

K(V, U)

^on Ev to Eu is a

03BB(P, ~)-nuclear

map. In

[1]

^it is shown that a l.c.s. E is

03BB(P, ~)-

nuclear if and

only

if for each U E

U (E)

there exists V E

6U(E)

^such

that

{03B4n (V, U)}n

^E

03BB (P, ~).

^We shall denote the class of

all 03BB (P, ~)-

nuclear _spaces

by N03BB(P,~).

A l.c.s. E is said to be stable if E x E is

isomorphic

^{to E. It is}

known that a nuclear

G--space

is stable if and

only

if for each _p E P there exists a q ^E P such that

{P2n/qn}n ~

^l~

[14].

^We say "P is a

stable G--set" to mean

that 03BB (P, ~)

^{is a stable}

G--space. Stability plays

^an

important

role in the

study

^{of various} permanence

properties of 03BB (P, ~)-nuclear

spaces; the

following

^{result can} be found in

[7;

Proposition 4.5].

1.2 PROPOSITION: Let P be a

countable,

monotone, ^nuclear

Gao-set,

P ⁼

{{pin}n :

^{i =} 1,

2, ...}.

^{Then the}

following

statements are

equivalent : (i)

^For

each j

^{E N} there exists a

03B3(j)

^{E N such that}

{Pi2n/Pyn(j)}n

^E

lao.

(ii) Ife k

^E

03BB(P, ~) for

^{each k ~ N and}

0:

^{N - N x N is a}

bijection defined by J3-1(k, ^m):

⁼

2 k-1 (2m - 1)

then there exist tk &#x3E; 0,

k E N,

such that the _sequence

{t03B21(n)03BE03B203B212(n)(n)}n ~ 03BB (P, ~)

^where

(3(n) = (J31(n), 03B22(n)).

(iii) If ç, 71

^E

À(P, ~) and

* 71: =

(Ç1, 711, e2e ~2, ...)

^{then there}

exists a

bijection

^{03C0: N ~ N such that}

{03B603C0(n)}n

^E

^{03BB (P, ~).}

(iv) N03BB(P,~)

^is closed under the

operation of forming

^countable

direct sums.

(v) N03BB(p,~)

^{is closed under the}

operation of forming finite

^Car-

tesian

products.

(vi) N03BB(P,~)

is closed under the

operation of forming arbitrary

Cartesian

products.

(vü)

^{The sum}

of

^two

03BB (P, ~)-nuclear

maps is a

À (P, ~)-nuclear

map.

(vili) 03BB (P, ~)

^{is stable.}

An operator ^{ideal A is said to be}

(i) surjective

^if for each closed

subspace

^{N of a Banach} space X and each T E

L(XIN, Y),

^{Y a Banach} space,

TQN

^E

A(X, Y) implies

T E

A (X/N, Y), QXN : ^{X ~ X/N}

denotes the canonical _{map onto}

X/N ;

(ü) injective

if for each closed

subspace

^{M of a Banach} space Y and each

T E L(X, M),

^{X a Banach} space,

JMT

^E

A(X, Y) implies

^{T E}

A(X, Y), J XM :

^{M ~ Y} denotes the

injection.

The

following

^result shows that the

operator

^{ideal of}

À(P, ~)-

nuclear

maps - P

^a

stable, countable,

monotone, nuclear G--set - is

injective

^and

surjective.

1.3 PROPOSITION: Let P be a

stable, countable,

monotone, nuclear

Gao-set,

^{X and Y be Banach} spaces with closed unit balls Ux and U,.

Then the

following

statements are

equivalent.

PROOF:

(i)

^~

(ii)

is shown in

[1]

^and

[7]. (ii)

~

(iii)

^and

(ii) ~ (iv)

are consequences of the facts that

Sngel(T) ~ s,,"(T)

^and

Snkol(T) ~ snpp(T)

for each n G N

[5]. (iii)

~

(ü)

^and

(iv) ~ (ii)

^are consequences of the facts that

Snapp(T) ~ (n

⁺

1) snel(T)

^and

Snapp(T) ~ (n

⁺

1)Snkol(T)

^for

each n E N

[5]

and the well known fact that

1(n

⁺

1)03BEn}n

^E

03BB (P, ~)

^for

lenln ~ 03BB (P, ~).

1.4 COROLLARY: Let P be a

stable, countable,

monotone, nuclear Gao-set. Then the operator ^ideal

of 03BB (P, 00 )-nuclear

maps is

injective

and

surjective.

PROOF: Let X and Y be Banach _{spaces, M} a closed

subspace

^of

Y,

^and

X/N

^a

quotient

space of X. Then

snel(T) = s’n e’(JM’T)

^{for all}

T E

L(X, M)

^and

Snkol(SQXN) = Snkol(S)

for all S ^E

L(X/N, Y) [5].

^Now the

proof easily

follows from

Proposition

^1.3.

1.5 COROLLARY: Let P be a

stable, countable,

monotone, nuclear

G--set,

^{X and Y be Banach} spaces such that each _map T E

L(X, Y)

is

À(P, ~)-nuclear.

^{Then X or Y must be}

finite-dimensional.

PROOF:

By Proposition

1.3 every

operator T E L(X, Y)

^{is of} type

03BB (P, ~),

^{hence of} type li.

By

^{a result of Pietsch}

[6],

^{X or Y must be} finite-dimensional.

REMARK: With methods used in

[11],

^{it can} be also shown that X or

Y is finite-dimensional if each nuclear _map

T E L(X, Y) is 03BB (P, ~)- nuclear,

^{P as in}

Corollary

^1.5.

2. On

03BB (~, P, ~)-Nuclearity

Throughout,

^let 03A6 denote the set of all

functions ~: [0, ~] ~ [0, ~]

which are

continuous, strictly increasing,

subadditive

with ~(0) = 0 and ~(1) = 1,

^and

satisfy

^the additional condition

(+)

^{there exist}

constants M ~ 1

and t~

^E

[0, ~]

^such

that ~ (~t) ~ ~~(Mt)

^{for t E}

[0, t~ ).

Note that all

examples given

ⁱⁿ

[10;2.6]

^{do have the additional}

property (+).

^{So far we} have been unable to find an

example

^{of a}

continuous, strictly increasing,

subadditive

function 0: [0, ~) ~ [0, ~)

with

~(0)

^{= 0} which does not fulfill condition

(+).

2.1 DEFINITION: Let

03BB (P, ~)

be nuclear.

For 0

^{EE 0 define the}

sequence space ^À

(~, P, ~) by

A l.c.s. E is called

À (4), P, ~)-nuclear

if for each U E

U(E)

^there

exists a V E

U(E)

^{such that}

{03B4n (V, U)}n

^E

À (0,

P,

~).

For ~

= id this definition _agrees with the definition of a

À(P, ~)-

nuclear _space, so we write

03BB (P, ~)-nuclear

instead of

03BB (id,

P,

~)-

nuclear.

We now show that the

concept

^{of À}

(~, P, 00 )-nuclearity

^is

exactly

the same as the concept ^of

À (Q, -)-nuclearity

^where

Q

^is

suitably

chosen

depending

^on

0.

PROOF:

(i)

^It is obvious that

Q

is countable and monotone and that for each _n, i E

Nqin

&#x3E; 0 and

qin ~ qin+1.

(ii)

^Given

q k, q 1~ Q,

^{we have to} show the existence of

a q

^{r ~}

Q

such that

q kql q r,

^{i. e.}

q knq ln~ Mq rn

^{for a constant M} &#x3E; 0 and each

n G N. Since P is a Kôthe set we find

p m

E P and M1 &#x3E; 0 such that

p kn ~M1pmn

^and

pln ~ M1pmn

^{for each n EN. Since}

03BB(P, ~)

^is

nuclear,

there exists a

p

^{G P with}

{pmn/psn}n

^{E 1,. This can be seen as follows.}

By

Lemma 1.1 there exists a

p’

^{E P with}

11 /p jn In

^E

l1;

^{we choose}

ph

^{E P}

with

p jn

^:5

M2p hn

^and

p Z = M2p n

^{f or n E} N, ^we also choose

p

^{s ~ P with}

(phn )2 ~ p n

^{for all n E N.}

(This

^{can be done}

by

^condition

(ii)

^{of a}

G~-set.)

From the

inequalities

we get

{pmn /psn }n

^{E 1,.} We also find an

integer

^{no such that}

pkn/psn ~

1,

p ln/p sn ~ 1,

^and

1 / p sn t~

^{for each n ~ no}

(here t~

is determined

by

condition

(+)).

^{Because of condition}

(+)

^{for the}

function 0

^{we then have}

for each n ~ no.

Again

^{because of the}

nuclearity

^of

03BB(P, ~)

^{and the} second additional condition for a Gao-set we find

p r E P,

n1 EN, n1 ~ no, such that

p snpsn ~ p rn

^for each n ~ n1. Therefore there exists K &#x3E; 0 such that

~-1(1/prn)~ K~-1(1/pkn)~-1(1/lnp)

for each n EN, ^i.e.

qkq1 qr.

(iii)

To prove the

nuclearity

^of

03BB(Q,~)

^{we use Lemma 1.1 and}

show the existence of a sequence

{qkn}n~Q

^such

that {1/qkn}n ~l1.

Since

03BB(P,~)

^{is nuclear we can find}

pk ~ P

^such

that {1/Pkn}n~l1.

Since ~

is subadditive we have

1/qkn = ~-1 (1/pkn) ~ M/pkn

^{for each}

n E N,

and 03BB (Q, ~)

is nuclear.

If 03BB (P, ~)

^{is a} power series _space of infinite type

then 03BB (Q, ~)

^{is in}

general

^{not a} power series _space as the

following example

^shows.

EXAMPLE: Define

pi

^{E P}

by pin:

⁼

ni,

n, i E N.

Let ~

^{~03A6 be the}

function

~log

^as

given

ⁱⁿ

[10],

^i.e.

sufficiently

^small

where a,

03B2,

03B3~^{0 are} chosen in such a way that

~log

is continuous

with ~(1) = 1. Then ~log~03A6

^and

~-1log(t) = exp (-03B1/t )

^for

tE(0,to].

Therefore

q in

⁼ exp

(ani)

^{for n ~ no, i E N, and}

03BB (Q, ~)

^{is not a} power series _space

[1].

The

following

lemma is well known and can be found in

[4].

2.3 LEMMA: Let P be a

countable,

monotone, nuclear Gao-set. Then

topologies

determined

by

^{the semi-norms}

Ui (x) : =

sup

lenlpn’,

^{i E N, are}

equivalent.

2.4 PROPOSITION: Let P be a

countable,

^nuclear Then the

following

statements are true.

equivalent.

(iii) 03BB (~, P, ~)

with the above

topologies

^is

topologically equivalent to 03BB(Q, ~).

PROOF:

(ii)

^{For x =}

{03BEn}n

^{E À}

(~, P, ~)

^we

always

^have

Úi(X):5 7Ti(X),

i ~N. On the other hand for i EN there exists

p

^{~P and}

p

^{l ~ P such}

that {1/prn}

^{E l and}

pipr pl;

^therefore

To prove

(i)

^and

(iii)

notice that because of Lemma

As an immediate consequence of

Proposition

^{4.6 in}

[7]

^we

finally

get

2.6 PROPOSITION: Let P be a

stable, countable,

monotone, ^nuclear Then the

following

statements are

equivalent.

(üi)

^{E is}

isomorphic

^{to a}

subspace of

^{a suitable}

I-fold product

3. On

03BB(~, P, N)-nuclearity

In this section we

study

^the concepts ^of

03BB(~, P, N)-nuclearity,

0

^E

0,

^and

03BB (P, N)-nuclearity,

^{P a}

countable,

^{monotone Goo-set. As in}

the case of

03BB(~, ^P, ~)-nuclearity

^we obtain that both concepts ^are

closely

related. The main result in this section is that whenever P is a

countable,

monotone,

stable,

nuclear Goo-set

03BB(P, ~)

^{is a} universal

generator

^{for the}

variety

^of

À(P, N)-nuclear

spaces. For

À(P, ~) = A-(a),

^{a a stable} exponent sequence, this result has been established in

[8].

Let P ⁼

{{pin}n:

^{i =} 1,

2, 3, . . .}

^{be a}

countable,

^{monotone Gao-set.}

Fix k E N. We define the sequence space

À(P, k) by

It is obvious that

03BB(P, k

⁺

1)

^C

03BB(P, k )

^and

03BB(P, (0) =

^{~ k}

03BBP, k).

^For

~~03A6

^define

03BB (~, P, k) by

^À

(~,

P,

k) : = {{03BEn}n: {~(|03BEn|)}n ^~03BB(P, k)l.

3.1 DEFINITION: Let P be a

countable,

monotone, ^nuclear

G--set,

~~

0. A l.c.s. E is said to be

(i) 03BB (~, P, k)-nuclear

^{if for each U E}

U (E)

^{there exists a V E}

U(E)

^{such that}

{03B4n ( V, U)}n~ 03BB(~, P, K);

(ii) À (0, P, N)-nuclear

^{if E is}

03BB(~,

^P,

k)-nuclear

^{for each k ~}

ko,

where

{1/pkno}n

^{E Il because of the}

nuclearity

^of

03BB (P, ~).

For 4&#x3E;

^{= id we}

write À(P, k)-nuclear resp. 03BB(P, N)-nuclear

^{instead of}

03BB(id,

P,

K)-nuclear

resp. ^À

(id, P, N)-nuclear.

If a is an exponent sequence and if P ⁼

{{k03B1~}n: k ~N},

^{then a l.c.s.}

E is

03BB (P, N)-nuclear

^{iff E is}

AN(03B1)-nuclear,

^i.e.

Ãk(03B1)-nuclear

^{for each}

k &#x3E; 1. So we indeed

generalize

^the concept ^of

AN(03B1)-nuclearity

^con-

sidered in

[8].

As in section 2 we now show that the concepts ^of

03BB (~ P, N)- nuclearity

^and

À(Q, N)-nuclearity

^{are the same if}

Q

^is

suitably

^chosen.

3.2 PROPOSITION: Let P be a

countable,

monotone, nuclear

G--set,

~

^{E 03A6.}

Define {qin}n by llqn’

^:=

~-1 (1lpin)

^and

Q

⁼

{{qin}n}.

^{Let E be a}

l.c.s. Then the

following

statements are

equivalent.

(i)

^{E is}

03BB (~, P, N)-nuclear.

(ii) E is 03BB (Q,N)-nuclear.

PROOF: Note that E is

03BB(~, ^P, N)-nuclear (resp. À(Q, N)-nuclear)

will follow if E is shown to be

03BB(~, ^P, k)-nuclear (resp. Ã(Q, k)- nuclear)

for each k ~

ko,

ko ^as in 3.1. Therefore we will show

(1) given

k ~ ko there exists l~ N such that

03BB (~, P, l + 1)~03BB (Q, k); (2) given

r ro there exists s E N such that

03BB (Q, s

⁺

1)

^C

À (0, P, r).

^Given

k, by

nuclearity

^of

03BB (Q, ~) (Lemma 2.2)

^{there exists 1 EN such that}

lq ’n/ q ’n 1.

^{E l1. If x =}

F-

^03BB

(~, P,

^{1 +}

1)

^then

~(l03BEnl)pln

^{~ 1 for n ~ ni}

therefore

~n |03BEn|qkn

^=~n

|03BEn|qknqln/qln

^{~. On theotherhand}

given

r,

by nuclearity

^of

03BB (P, ~),

^{find sEN} such that

{Prn/Psn}n

^{E Il. So if}

x = {03BEn}n ~03BB(Q, s + 1)

^we

have |03BEn| ~1/qsn

^{for n~ ns} and therefore

~n ~(|03BEn|)prn

⁼

~n ~(~03BEn)psnprn/Psn

^00; this shows

(1)

^and

(2).

To _prove

(i) ~ (ii)

^{fix k ~}

ko,

^{find l E N such that}

03BB(~, P,

^{1 +}

1)

^C

03BB(Q, k).

^{Since E is}

03BB (~, P, N)-nuclear,

^E

is 03BB(~, P,

^{l +}

1)-nuclear

^and therefore

03BB(Q, N)-nuclear.

^The

implication (ii)

~

(i)

^{can be}

proved

ⁱⁿ

the same way.

REMARK: It is

easily

^{seen that}

for ~~03A6 03BB(~, P, N)-nuclearity implies ~-nuclearity [9].

^{But the reverse of this}

implication

^{is not true}

in

general

^{as the}

following example

^shows.

Take ~ = ~log;

consider the _power series _space

A-(a)

^with

03B1n:= (n + 1)2.

^We show that

Aao(a)

^is

~log-nuclear

^{but not}

À

(4)log, ^R, N)-nuclear

^{if R : =}

11(n

⁺

1)k}n :

^{k =}

1,2, . . .}.

^The

~log- nuclearity

^of

Aoo(a) immediately

follows from Korollar 3.6 in

[9].

^But

A-(a)

^is

03BB(~log, R, N)-nuclear

^{if and}

only

^{if there exists M} &#x3E; 1 such that

{~log(1/M03B1n )}n ~03BB(R, k)

^{for each k}

[8].

^Therefore

A~(03B1 )

^is

03BB (~log, R, N )-nuclear

^{if and}

only

if for each k~ko the sum

In

(n

⁺

1)k/03B1n log

^{M =} En

(n

⁺

1)k-2/log

^{M is finite which}

obviously

^is

not true.

In order to answer the

question

^{what the model of a universal}

03BB (~, P, N)-nuclear

space

is,

^{it is}

enough

^to describe the model of a

universal

03BB (Q, N)-nuclear

space; ^so from now on P is

always

sup-

posed

^{to be a}

countable,

monotone, nuclear G-set.

Since

03BB (P, N)-nuclearity implies nuclearity

^one

easily

obtains fol-

lowing

permanence

properties.

3.3 PROPOSITION:

Subspaces, quotient

^spaces

by

^closed

subspaces,

and

completions of 03BB (P, N)-nuclear

spaces

are 03BB(P, N)-nuclear.

3.4 LEMMA: For each k ~ N there exists ^{r E} N such that

1(n

⁺

1))n)n

^E

03BB(P, k)

^whenever

lenln ~03BB (P,

^{r +}

1).

PROOF : Fix k E N. Since

À(P, ~)

^{is nuclear} there exists 1 E N such that

{pkn/pln}n

^{E Il.} Therefore for each n E N

(pko/lo)

^{+... +}

(pkn/pln ) ~

M

and n + 1 ~ p lnM/pko = : M1pkln.

^This

implies

3.5 LEMMA: Let

Hl,

H2 be Hilbert spaces and T E

L(Hl, H2).

^Then

(i) given k,

^{T is}

À(P, k)-nuclear if

^{T is}

of

type

À(P, k);

(ii) given k,

there exists r so that T is

of

type

03BB(P, k) if

^{T is}

À(P,

^{r +}

1)-nuclear.

PROOF:

(i):

^{We recall that for a} compact operator ^{T E}

L(Hi, H2)

^T

has a

representation

^{Tx =}

~n 03BBn (x, en)fn

for suitable orthonormal sequences

lenl, {fn}

^{in Hl} resp.

H2;

also An ⁼

8n(TUHD UH2)

⁼

snpp(T).

(ii)

^{Fix k. Lemma 3.4} guarantees the existence of ^r E N such that

1(n

⁺

1))n)n

^E

03BB(P, k)

^if

lenl,,

^E

03BB (P, r

⁺

1).

^{Now let T be}

03BB(P,

^{r +}

1)- nuclear,

^{then T has a}

representation

^{Tx =}

~n 03B3n~x, an~03B3n, {03B3n}n

^E

03BB(P, r + 1), ~an~=~yn~=1,an ~H’1 yn~H’2. Since Sappn(T)~~~i=n 03B3i, we

have

3.6 PROPOSITION: Let P be a

countable,

monotone,

stable,

^nuclear

G--set. Then countable direct sums

of 03BB(P, N)-nuclear

spaces ^are

À (P, N)-nuclear.

PROOF: We

only

^{indicate a}

partial proof

^{and refer} the reader to

[1,

Theorem

2.8]

^{for the rest of the}

proof.

In E ⁼

~~i=1

^{Ei a}

typical

fundamental system ^of

neighbourhoods

^of

0 is of the form U ⁼

F((Ui)i)

^{where each Uk is an}

absolutely

convex,

closed,

^and

absorbing neighbourhood

^{of 0 in Ek and r} represents ^the closed convex hull of the union.

Stability

^of

03BB (P, ~)

^now

gives

^a

single valued, increasing

map y : ^N~N such that

03B3(j) &#x3E; j

^and

p2n

⁼

0(Pyn(j)).

^Then

We have to show that E ⁼

~i

^{Ei is}

À(P, k)-nuclear

for each k - ko.

Fix k,

^{assume i k.}

Using

the fact that Ei

is À(P,N)-nuclear

^and

therefore nuclear for each

neighbourhood Ui ~U(Ei)

^{we find a}

neighbourhood Wi ~ U (Ei)

^{so that the canonical map Ki :}

^{(Ei) w,}

^~

(Ei)ul

^is

represented by

i ^2:: k we get ^a

representation

Thus we have shown that

{t03B2(n)03BE03B2(n)03B2(n)}~03BB(P,k).

As in the

proof

^of

Theorem 2.8 in

[1],

we now can show that E is

03BB (P, k)-nuclear

^and

since this can be done for

each k ~ ko,

^{E is}

03BB (P, N)-nuclear.

3.7 COROLLARY:

Arbitrary products of 03BB(P, N)-nuclear

spaces

again

^are

À(P, N)-nuclear

^{whenever P is a}

countable,

monotone,

stable,

^{nuclear G--set.}

3.8 PROPOSITION: Let P be a

countable,

monotone, nuclear G--set.

Then

03BB(P,~) is 03BB(P,N)-nuclear.

PROOF:

Since 03BB(P,~)

^{is a nuclear}

G--space,

there exists a

p’

^{i ~ P}

such

that {1/pin}~l1. Given k,

rEN, there exists a

p

t E P such that

prpk pt ;

^{we also find a}

p

^{s ~ P so that}

p tp i ps.

^{We then have}

so

03BB (P,~)

^is

À(P,k)-nuclear

for each k.

Our next result shows that

03BB(P, ~)

^{is a universal} generator ^{for the}

variety

^of

A (P, N)-nuclear

spaces if P is stable.

3.9 PROPOSITION: Let P be a

countable,

monotone,

stable,

^nuclear

G--set;

^{then each}

À(P, N)-nuclear

space E is

isomorphic to a subspace

of [À(P, oo)]’ for

^a suitable I.

PROOF: Let kEN be fixed.

By stability

let y : ^N~N be a

single valued, increasing

function such that

y(j) &#x3E; j

^and

p2n

⁼

0(pyn(j)).

^Write

k : = 03B3k (k).

^{Since E is}

^À(P, N)-nuclear

^{we have}

pk

^E

L1(E),

^{and there} exists an

absolutely

convex,

closed,

^and

absorbing neighbourhood

U E

6U (E)

^{so that} Euo

( U°

^{is the}

polar

^of

U)

^{is a Hilbert} space; ^now

by Proposition

^{IV.1 of}

[12]

there exists an orthonormal basis

{ekn~n

ⁱⁿ

Euo ^{so that the} set

is

equicontinuous

^{in E’.}

Rearrange

^{the set}

{ekn: k, n =1, 2, . . .}

into ^a

single

sequence

by using

^the

bijection

map 03B2 : ^{N ~ N x N} with

03B2-1(k,n): =2k-1(2n-1);

^{use the} Gram-Schmidt _process to

obtain a new orthonormal basis

{em~

^for E’uo. We then have

em =

~n~m/2k (em, ekn)ekn

and therefore

Since

p;km:5 Mkp03B3km(k)

⁼

Mkpkm

^and since without loss of

generality

^we

may ^assume Mk =

1,

^we get

and therefore

pkmek ~ Ak.

So we have shown that there exists an orthonormal basis

{em}

^of

Euo

^{such that}

lp lem :

^{m =}

1, 2, . . .}

^is

equicontinuous

ⁱⁿ

Eue,

^{for each} fixed k.

Let 6U ⁼

(Ui :

^{i E}

I )

^{be a base of}

neighbourhoods

^{of 0 in E so that}

each Ui is

absolutely

convex,

closed,

^and

absorbing

^and

Eu?

^{is a} Hilbert space. For each i ~ I we can get ^an orthonormal basis

(ei :

m ⁼

1, 2, . . .}

^of

Eu?

such that the sets Bi,k : ^:=

lp lei :

^{m =} 1,

2, . . .}

^are

equicontinuous

^for each fixed

k ;

^{for each}

i E I,

define the _map Ti :

E~03BB(P,~) by Tix :={~x, eim~}m, ; Ti

^{goes into}

À(P,oo)

^{and is con-} tinuous. Define T:

E ~[03BB(P, ~)]I by Tx :={Tix}i.

^{Then T is con-}

tinuous and one-to-one. With obvious

changes,

^{the rest of the}

proof

^is

exactly

^{the same as of}

Proposition

^{3.4 in}

[8].

In

[2],

Fenske and Schock consider the class f2 of all l.c.s. E such that the set w of all

strictly positive, non-decreasing

sequences of reals is contained in

0394 (E). They prove f2

^{to be a}

stability

^{class of}

nuclear _spaces and therefore closed under the

operations

^of

forming

completions, subspaces, quotients by

^closed

subspaces, arbitrary

products,

^{countable direct} sums, tensor

products,

^and

isomorphic

images;

^more

precisely they

^{show D =}

~~~~N~

^where

N~

^{denotes the} class of

~-nuclear

spaces. The

following proposition

^{extends this}

result.

3.10 PROPOSITION: Let P be a

countable,

monotone,

stable,

^nuclear Gao-set. Then D ⁼

~~~03A6N03BB(~,P,N).

PROOF: Let E be a 1.c.s. in

~~~03A6N03BB(~,P,N)

^{then E E}

~~~03A6N~.

^If

03A6

denotes the set of all

continuous, strictly increasing,

subadditive

functions ~ with ~(0)=0,

^then

obviously ~~~03A6N~=~~~03A6N~.

Therefore

~ ~~03A6 N03BB(~,P,N) ~ 03A9.

^{Now fix E in}

03A9, then 03C9 ~ ~ (E).

^Take

k ~ N, ~ ~ 03A6

^then

(1/~ -1 (1/pkn )}

^{E w, so}

given

^{U e}

U (E )

^one may find Vk ^E

U (E)

^{such that}

8n (Vk, U)/~ -1(1/pkn) ~ 0

for n ~ ~. For n ~ no we

then have

~ (03B4n (Vk, U))pkn ~ 1,

^{and E is}

03BB(~, P, N)-nuclear

^{for each}

cp

^{~ 03A6.}

REFERENCES

[1] E. DUBINSKY and M. S. RAMANUJAN: On 03BB-nuclearity. ^{Mem. Amer. Math. Soc.}

128 (1972).

[2] CH. FENSKE and E. SCHOCK: Nuclear spaces of maximal diametral dimension.

Compositio ^{Math. 26} (1973) ^303-308.

[3] ^{G. KÖTHE:} Topologische ^{lineare Räume I.} Berlin-Göttingen-Heidelberg ^1960.

[4] A. PIETSCH: Nukleare lokalkonvexe Räume. Berlin 1969.

[5] A. PIETSCH: s-numbers of operators in Banach _spaces. Studia Math. 51 (1974) 199-221.

[6] ^{A. PIETSCH:} Small ideals of operators. Studia Math. 51 (1974) 265-267.

[7] ^M. S. RAMANUJAN and T. TERZIOGLU: Diametral dimension of cartesian products, stability ^{of smooth} sequence spaces and applications. ^{J. Reine} Angew.

Math. (to appear).

[8] M. S. RAMANUJAN and T. TERZIOGLU: Power series _space 039Bk(03B1) ^{of finite} type and related nuclearities. Studia Math. 53 (1975) ^1-13.

[9] B, ROSENBERGER: 03A6-nukleare ^Räume. Math. Nachrichten 52 (1972) ^147-160.

[10] B. ROSENBERGER: Universal generators for varieties of nuclear spaces. Trans.

Amer. Math. Soc. 184 (1973) ^275-290.

[11] ^B. ROSENBERGER: Approximationszahlen, p-nukleare Operatoren ^{und Hil-} bertraumcharakterisierungen. Math. Annalen 213 (1975) ^211-221.

[12] T. TERZIOGLU: Die diametrale Dimension von lokalkonvexen Räumen. Collect.

Math. 20 (1969) ^49-99.

[13] T. TERZIOGLU: Smooth sequence spaces and associated nuclearity. Proc. Amer.

Math. Soc. 37 (1973) ^497-504.

[14] T. TERZIOGLU: Stability ^{of smooth} sequence spaces. J. Reine Angew. ^Math. (to appear).

(Oblatum 24-XII-1974 M. S. Ramanujan ^B. Rosenberger

&#x26; 1-IX-1975) Department of Mathematics Fachbereich Mathematik

University ^of Michigan Universitât Kaiserslautern Ann Arbor, Michigan ^{48104 D} 6750 Kaiserslautern

Pfaffenbergstral3e ⁹⁵